The Fourier transform of the zeroth-order Bessel function J0(x) is definitively a rectangular function (rect function), as stated in the Wikipedia article on Fourier transforms. However, in the context of optics, the Fourier transform of a ring disk or ring slit is also represented by the zeroth-order Bessel function. The distinction lies in the dimensionality of the transforms, with the rect function applicable in Cartesian coordinates and the Bessel function in radial coordinates. The two representations are interconnected, with the Fourier transform of a ring disk yielding a rectangular window divided by sqrt(1-omega^2).
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KFC said:In wiki (http://en.wikipedia.org/wiki/Fourier_transform), there said the Fourier transform of the Bessel function (zeroth order J0) is a rect function (window). But I also saw a text (about optics) that the Fourier transform on a ring slit (or ring disk) is zeroth-order Bessel function, so which one is correct? If wiki is correct, what is the Fourier transform on a ring disk?